1. Field of the Invention
The present invention generally relates to a spectacle (or eyeglass) lens having a pair of first and second refractive (or refracting) surfaces (hereunder sometimes referred to simply as first and second surfaces, respectively), one or both of which have aspherical shapes. More particularly, the present invention relates to an astigmatic spectacle-lens (namely, a spectacle lens for correction of astigmatism).
2. Description of the Related Art
It is first supposed that the shape of a curved surface of a spectacle lens is represented by coordinates of a coordinate system shown in FIG. 1. Namely, it is supposed that an optical axis, thereof is set as the x-axis of the coordinate system shown in this figure and that the orientation (namely, the positive direction) of the optical axis (or x-axis) is the horizontally rightward direction thereof (as viewed in this figure). In this case, the eye of a spectacle lens wearer (namely, the wearer's line of sight) is directed to the (horizontally) leftward direction. Moreover, the center O' of rotation (or rotation center O') of his or her eye is positioned on the x-axis. Furthermore, the y-axis assumes the vertically upward direction and the z-axis takes a direction determined according to the right-hand rule. Additionally, it is supposed that the origin O of the coordinate system is a point of intersection of the optical axis and a refracting surface. Especially, in the case that no decentration occurs in the spectacle lens, the x-axis coincides with a normal to the refracting surface at the origin thereof.
Astigmatic spectacle-lens is required to employ one or two refracting surfaces (hereunder sometimes referred to as astigmatic surfaces), whose curvatures vary with directions, as one or both of the first and second surfaces thereof. Astigmatic surface is a curved surface whose curvature (namely, a normal curvature) at the origin of a curve (namely, an orthogonal curve or trajectory), which intersects with a plane containing the x-axis (namely, an orthogonal cross-sectional plane) and with the curved surface, changes depending upon an angle .theta. of intersection of the orthogonal cross-section plane and an x-y plane. According to differential geometry, a normal curvature C(.theta.) is calculated from two principal curvatures (namely, maximum and minimum curvatures of orthogonal trajectories or curves on orthogonal cross-section planes intersecting with each other at right angles) in accordance with the following equation: EQU C.sub.y cos .sup.2 .theta.+C.sub.2 sin .sup.2 .theta.
Incidentally, it is supposed that the directions corresponding to these principal curvatures are the y-direction and z-direction, respectively.
Hitherto, a toric surface shown in FIG. 2 has been employed as such an astigmatic surface. Toric surface is a curved surface generated by rotating a curve (namely, a generating line) x=f(y), which extends in the x-y plane, around a line (x=1/C.sub.z and z=0), which is employed as an axis of rotation, or is curved surface generated by rotating a curve x=f(z), which extends in an x-z plane, around a line (x=1/c.sub.y and y=0), which is employed as an axis of rotation. Incidentally, the curvature of the curve x=f(y) in the case where y =0 is c.sub.y, and similarly, the curvature of the curve x=f(z) in the case where z=0 is C.sub.z. Hitherto, mostly, the curves x=f(y) or x=f(z) are circles. Thus, when the principal curvatures C.sub.y and C.sub.z are determined, there are only the following two kinds of refractive surfaces: a curved surface generated by rotating a curve ##EQU2## which extends in the x-y plane, around the line x=1/C.sub.z and z=0, which is employed as an axis of rotation; and a curved surface generated by rotating a curve ##EQU3## which extends in the x-z plane, around the line x=1/C.sub.y and y=0, which is employed as an axis of rotation. In the case that C.sub.y &lt;C.sub.z, the former curved surface is of the barrel type (see FIG. 2); and the latter curved surface is of the tire type (see FIG. 3).
In the case of an aspherical spectacle-lens consisting of a spherical surface and a toric surface whose generatix (namely, generating line) is a circle, it is necessary for reducing a residual astigmatism and a mean power error (to be described later) to select a lens, which has a (relatively) large curvature, at the time of designing thereof. As a result, this aspherical lens is thick. This is undesirable from the viewpoint of the beauty of the external appearance thereof.
There has been proposed a technique of forming an axisymmetric (or axially symmetric) surface in the shape of an aspherical surface so as to reduce the astigmatism, thickness and weight of a spectacle lens. For example, in Japanese Unexamined Patent Publication No. 64-40926/1989 Official Gazette, an axisymmetric system is expressed by the following equation: ##EQU4## This is a typical equation that represents an aspherical surface and is effective in the case of an axisymmetric lens but has only a limited effect in the case of an astigmatic lens.
The residual astigmatism and the mean power error (to be described later), which correspond to a direction corresponding to one of the principal curvatures, can be corrected by employing a free curve as the generating line of the toric surface instead of a circle to thereby increase the degree of freedom. However, the orthogonal curved, which corresponds to a direction corresponding to the other principal curvature, is a circle. Thus, a curve having a (relatively) large curvature should be selected so as to correct the residual astigmatism and the mean power error. Although there has been proposed an idea of forming the former refractive surface (namely, the axisymmetric surface) in the shape of an aspherical surface in this case (namely, a method of forming each of the refractive surfaces of the lens in the shape of an aspherical surface), it is difficult to machine each of the refractive surfaces of the lens in such a manner. Additionally, there has been proposed a method of forming each of the refractive surfaces of the lens in the shape of a toric surface (see Japanese Unexamined Patent Publication No. 54-131950 Official Gazette).
To correct the aberration of the lens simultaneously with the correction of astigmatism thereof by a refractive surface thereof, there should be achieved a breakthrough in overcoming difficulties due to the idea of utilization of a toric surface which is a surface of revolution. Namely, there should be utilized a (curved) surface adapted so that not only orthogonal trajectories or curves respectively corresponding to the directions, which respectively correspond to both of the principal curvatures, but also orthogonal trajectories respectively corresponding to other directions are free curves. Prior art techniques for generating such a curved surface are disclosed in Japanese Examined Patent Publication No. 47-23943/1972 Official Gazette, Japanese Unexamined Patent Publication No. 57-10112/1982 Official Gazette and (PCT) International Publication No. WO93/07525 Official Gazette. However, in Japanese Examined Patent Publication No. 47-23943/1972 Official Gazette and Japanese Unexamined Patent Publication No. 57-10112/1982 Official Gazette, no strict mathematic representations of such a free (astigmatic) surface are defined. Thus, such a free surface cannot be realized. Further, in International Publication No. WO93/07525 Official Gazette, such a free astigmatic surface is represented by the following equation (containing variables that are consistent with those of a coordinate system shown in FIG. 1) and equations derived therefrom: ##EQU5## where r.sup.2 =y.sup.2 +z.sup.2 ; and A.sub.n . . . m,j denotes predetermined coefficients respectively corresponding to items.
The herein-above described equation is obtained by extending the previously described equation, which represents an axisymmetric surface, in such a manner as to be a two-dimensional one. Free curved surface, which is represented by this equation, assures the continuity of a differential of arbitrary order and meets requirements for correcting the astigmatic conditions of the central portion of the lens, However, the excellent analyzability of this two-dimensional equation causes demerits in designing a lens. For instance, it is necessary for improving aberration conditions at a certain place or portion to change all of the coefficients in this equation. Further, in the process of changing such coefficients, owing to the properties of finite power series, the curved surface is liable to undulate (namely, what is called Runge phenomenon is to occur). Thus, it is difficult to make the power series converge. The degree of difficulty in converging the power series becomes higher with increasing the number of terms of the power series (n, m, l) so as to raise the degree of freedom, Although such difficulty can be alleviated to a certain extend by taking measures, for example, limiting the numerical quantities (or classes) of the coefficients, this does not reach a significant solution.
The present invention is accomplished in view of the aforementioned background.
Accordingly, an object of the present invention is to provide an aspherical spectacle-lens that can minimize the residual astigmatism and the aberration thereof in all of third eye positions (namely, tertiary positions) and can reduce the center thickness and edge thickness thereof.